Convex pentagons that admit $i$-block transitive tilings
| Autor: | Casey Mann, David Von Derau, Jennifer McLoud-Mann |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2015 |
| Předmět: |
Transitive relation
Hyperbolic geometry 010102 general mathematics Regular polygon Block (permutation group theory) Metric Geometry (math.MG) Basis (universal algebra) Algebraic geometry Type (model theory) 01 natural sciences 010309 optics Combinatorics Pentagon Mathematics - Metric Geometry 0103 physical sciences FOS: Mathematics Geometry and Topology Hardware_ARITHMETICANDLOGICSTRUCTURES 0101 mathematics Mathematics |
| Popis: | The problem of classifying the convex pentagons that admit tilings of the plane is a long-standing unsolved problem. Previous to this article, there were 14 known distinct kinds of convex pentagons that admit tilings of the plane. Five of these types admit tile-transitive tilings (i.e. there is a single transitivity class with respect to the symmetry group of the tiling). The remaining 9 types do not admit tile-transitive tilings, but do admit either 2-block transitive tilings or 3-block transitive tilings; these are tilings comprised of clusters of 2 or 3 pentagons such that these clusters form tile-2-transitive or tile-3-transitive tilings. In this article, we present some combinatorial results concerning pentagons that admit i-block transitive tilings for $$i \in \mathbb {N}$$ . These results form the basis for an automated approach to finding all pentagons that admit i-block transitive tilings for each $$i \in \mathbb {N}$$ . We will present the methods of this algorithm and the results of the computer searches so far, which includes a complete classification of all pentagons admitting i-block transitive tilings for $$i \le 4$$ , among which is a new 15th type of convex pentagon that admits a tile-3-transitive tiling. |
| Databáze: | OpenAIRE |
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